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 A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4. 6
 5, 13, 89, 389, 2593, 11593, 11593, 11593, 11593, 373649, 766261, 3358169, 12204889, 12270077, 12270077, 12270077, 297387757, 297779117, 297779117, 1113443017, 1113443017, 1113443017, 1113443017, 1113443017, 84676452781, 84676452781, 689101181569, 689101181569, 689101181569, 3278744415797, 3278744415797, 3278744415797, 3278744415797 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, A4. David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 163. LINKS Table of n, a(n) for n=1..33. J. K. Andersen, Consecutive Congruent Primes. D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689] EXAMPLE a(9) = 11593 because "[t]his number is the first in a sequence of 9 consecutive primes all of the form 4n + 1." MATHEMATICA NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 19} ] CROSSREFS Cf. A057619, A057620, A057622, A055623, A055624. Sequence in context: A263468 A350467 A081560 * A092567 A055623 A280294 Adjacent sequences: A057621 A057622 A057623 * A057625 A057626 A057627 KEYWORD nonn AUTHOR Robert G. Wilson v, Oct 09 2000 EXTENSIONS More terms from Don Reble, Nov 16 2003 More terms from Jens Kruse Andersen, May 29 2006 STATUS approved

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Last modified April 19 09:19 EDT 2024. Contains 371782 sequences. (Running on oeis4.)